Evaluation of thermal conductivity models and dielectric properties in metal oxide-filled poly(butylene succinate-co-adipate) composites

This study examines how various nanofillers impact thermal conductivity, dielectric characteristics, and electromagnetic interference (EMI) shielding potential of bio-based and biodegradable poly(butylene succinate-co-adipate) (PBSA). TiO2, NiFe2O4, Fe2O3, and Fe3O4 were selected as fillers for nanocomposites at 4–50 vol.% (12–81 wt.%). The nanocomposites were analyzed in three domains: structural (scanning electron microscopy, energy dispersive X-ray spectroscopy mapping, density, tensile testing), thermal (light flash analysis, literature models), and dielectric (AC conductivity, permittivity, EM shielding effectiveness (SE)). The investigated fillers showed good dispersion and compatibility with the PBSA matrix. LFA was analyzed according to literature models, where Bruggeman and Agari models showed the best fit at high concentrations. The dielectric analysis revealed that most of the nanocomposites did not reach percolation; thus, producing thermally conductive plastics that are electrically insulating. EMI shielding was limited to frequencies below 10 Hz, with the notable exception of Fe3O4 (100 nm and loading of > 25 vol.%), which showed shielding at frequencies up to 105 Hz. The investigated composites based on a biodegradable polyester and abundant metal oxide nanofillers are suitable for the production of cheap, ecological, and electrically insulating heat dissipation solutions required for modern and lightweight applications.

In the dynamic and rapidly evolving landscape of modern technology, particularly in the context of new advances in microelectronics, 5G communication equipment, lightweight aerospace, and electric transportation technologies, there has been a significant drive towards higher frequency, power, and transistor density in electronic systems and components.The trend towards miniaturization and growing power densities, as well as a push for energy-efficiency in electronic devices, requires new solutions for efficient heat dissipation for longterm operation 1 .A broad understanding of thermally conductive materials is necessary to achieve the desired goal, which requires engineering parameters, models, and their verification.
The development of thermally conductive materials that are electrically insulating, such as polymer composites, has become increasingly important in materials science 2 .Thermally conductive insulators find use in heat dissipation applications where electrical conductivity might cause short-circuits, electromagnetic non-compliance, or other undesirable effects [3][4][5] .In the applications where electrical conductivity is permissible, conductive fillers can impart composites with electromagnetic (EM) shielding, protecting against electromagnetic interference (EMI) and electrical noise, thus improving the operational stability of devices 6 .
Polymers, traditionally favored for their excellent flexibility, low density, corrosion resistance, and energysaving low-temperature processability, encounter limitations due to their inherently low dielectric constants and thermal conductivity.This has led to a surge in interest in organic-inorganic hybrid composite strategies, particularly focusing on the integration of thermally conductive fillers in the polymer matrix.The literature shows that for applications focused on heat dissipation, a mix of fossil-based commodity and engineering plastics have been studied, e.g., polypropylene 7 , polyamide-6 8 , low density polyethylene 9 , polyvinylidene fluoride 10 .New emerging bio-based and biodegradable matrix solutions have been underrepresented in the thermally conductive

Composite preparation
The composites were prepared using a simple solvent-based method.First, the fillers were dispersed in chloroform using sonication for 15 min (Hielscher UIS250V, Hielscher Ultrasonics GmbH, Teltow, Germany).Afterwards, the nanoparticle suspension was combined with a low concentration solution of PBSA in chloroform, and homogenized via sonication (15 min), followed by high-shear mixing at 10000 RPM for 30 min (Silverson L5M-A, Silverson Machines Ltd., Chesham, United Kingdom).The polymer-nanoparticle solution was then cast into a Petri dish and left overnight in a fume hood.Finally, the polymer film was further desiccated in a vacuum drying oven for 24 h at 70 °C.The filler loading values were chosen according to experiment design, to find critical values and evaluate filler dispersion and distribution (Table 1).The solvent cast composites were formed into 0.7 and 0.45 mm thick plates using compression molding at 135 °C for 3 min (Carver Inc., Wabash, IN, USA) followed by rapid cooling between steel plates at room temperature.Some filler was lost as particles separated from the cast polymer films.To ensure adequate analysis, filler volume percentages were recalculated by extrapolation from measured density values.To reflect the real filler percentage, samples were abbreviated from extrapolated vol.% (Table 1).Some of the highly loaded composites were too brittle for processing samples to precise shapes required for the specific measurements.www.nature.com/scientificreports/ The nanocomposite density was measured using hydrostatic weighing at room temperature.The sample weight was measured in air and in ethanol at room temperature on analytical scales Sartorius KBBA 100 with a YDK 01 density measurement kit (Sartorius AG, Göttingen, Germany), and the sample density was calculated according to the equation specified by the manufacturer: where W a (g) is the sample's weight in air, W fl is the sample's weight measured submerged in ethanol (g), ρ fl is the density of used ethanol (0.805 g•cm −3 ), which was determined with a hydrometer, and ρ a is the density of air (0.0012 g•cm −3 ).
Tensile testing was carried out at room temperature using a universal testing machine, the Tinius Olsen 25ST (Horsham, PA, USA), equipped with a 5 kN load cell at a speed of 1 mm per minute up to 2% strain, followed by 2 mm per minute until specimen fracture.The dog-bone shaped specimens with a gauge length of 21 mm, a width of 5 mm, and a thickness of 0.4 mm were cut out of compression-molded composite films.

• Thermal characterization
The thermal dissipative properties of the composites were obtained using light flash analysis apparatus LFA 447 NanoFlash (NETZSCH-Gerätebau GmbH, Selb, Germany) equipped with a standard 12.7 mm sample holder for through-plane diffusivity measurements according to ISO 22007-4.Square samples were coated with a graphite-based coating Graphit 33 as per manufacturer recommendations, to ensure equal absorbance and comparability with the reference samples.The thermal conductivity is calculated according to the equation: where is thermal conductivity (Wm −1 K −1 ), α is diffusivity (mm 2 s −1 ), ρ is density (g•cm −3 ) C p is specific heat capacity (Jg −1 K −1 ).
LFA measures sample diffusivity according to the half-time method: where d is the sample thickness, and t 1 2 is the half-rise time of the IR detector signal.As polymer composites are too insulating to use the adiabatic equation, the Cowan model 22 provided by the LFA analysis software suite was determined as the most accurate.Sample specific heat capacities were determined using the LFA analysis software suite by comparison to a reference with a known heat capacity.As is temperature dependent, measurements were carried out at three temperatures: 25, 35, 45 °C.Each sample was subjected to 5 consecutive measurements with a 120 s delay to allow the samples to return to thermal equilibrium.The measurement standard deviation is within 0.02 Wm −1 K -1 .
To model the thermal conductivity of the composites, we applied several models from the literature (Table 3).
To determine the thermal activation energy or the thermal dependence of thermal conductivity, a modified Arrhenius equation 23 was used: where 0 is the extrapolated inherent thermal conductivity at infinite temperature (Wm −1 K −1 ), E a is the energy of thermal activation, T is absolute temperature (K), and k is the Boltzmann constant (8.617•10 −5 eVK −1 ).In this case E a is calculated by taking the slope of the Arrhenius plot.It must be noted that thermal conductivity activation energies are valid only within the bounds of the tested temperature interval.

• Electric-dielectric characterization
The electric and dielectric properties-AC conductivity and dielectric permittivity-were characterized using a broadband dielectric spectroscope Novocontrol BDS 50 (Novocontrol Technologies GmbH and Co. KG, Montabaur, Germany).Composites were cut into 30 mm disks and placed between plate electrodes of the device.The measurements were carried out at frequencies from 10 −2 to 4 × 10 7 Hz at room temperature.
Dielectric EMI shielding efficiency was calculated over the tested frequency from the real and imaginary dielectric components using the following equations 24 : (1) where Ŵ is the reflectance, T is the transmittance, f is the frequency, d is the sample thickness, c is the speed of light, ε′ − jε ′′ is the complex permittivity, µ′ − jµ ′′ is the complex permeability (taken to be 1 when evaluating only dielectric effects), SE is the shielding efficiency (dB).
In the microwave frequency range from 25 to 40 GHz, a custom-made (Faculty of Physics, University of Vilnius, Lithuania) thin rod waveguide spectrometer as described in the reference 25 was used.Disc-shaped compression-molded specimens with a thickness of 0.45 mm were placed in the waveguide holder.The electromagnetic shielding properties of the nanocomposites in the tested range were calculated according to equations 26,27 : where k z = 2π and k 2z = 2πε 2 are wave numbers in the vacuum and the sample's media correspondingly, and τ is the thickness of the layer.The absorption of the layer was calculated according to:

Structure and morphology
During the processing stage, some filler was lost as particles separated from the cast polymer films.As a result, it became necessary to conduct density measurements to ascertain the volume concentration of filler that remained.True volume fraction was extrapolated from the density measurements ( ρ EX ) and rule-of-mixtures ( ρ th ) assuming that the samples have no voids present in the structure.The composites filled with 100 nm particles exhibited the smallest volumetric deviation ( V p = ρ th ρ EX % ), with the extent of this deviation varying according to the average particle size.For fillers with a size below 50 nm, the non-magnetic TiO 2 displayed the least deviation.Considering both aspects, we suggest that the compatibility between the filler and the matrix in our composite is influenced by two main factors.Firstly, larger particles possess a greater surface area, which allows for the creation of a higher number of adhesive bonds.Secondly, during the solvent casting process, small magnetic particles tend to stick together weakly through dipole-dipole interactions, which in turn hinders their ability to adhere to the matrix.The surface energy of nanoparticles also varies with particle size 28 .However, this is unlikely to significantly affect the wettability of the filler, due to the relatively large size of our particles.
The dispersion and distribution of nanofillers in composites filled with Fe 2 O 3 and Fe 3 O 4 (100 nm) were qualitatively evaluated using SEM (Fig. 2).This assessment focused on Fe 2 O 3 and Fe 3 O 4 (100 nm) composites with filler concentrations of 4, 28/25, and 46/40 vol.%, which have the highest thermal conductivities among selected fillers.At the lowest concentrations, Fe 3 O 4 particles tended to agglomerate in uniform clusters with a size under 1 μm (Fig. 2d).In contrast, Fe 2 O 3 showed better dispersion and smaller clusters or no agglomeration (Fig. 2a).Overall, both fillers show uniform distribution and good dispersion within the composite.As shown in our previous study 14 , Fe 3 O 4 particles are likely to re-agglomerate due to magnetic interaction after sonication and homogenization.This impacts particle dispersion and promotes formation of clusters 29 .The clusters are small enough to not get destroyed during compression molding.
With an increase in the filler concentration (28 vol.%),Fe 2 O 3 composite had a much more densely packed structure with filler packed in small clusters (Fig. 2b).These clusters are also visible at fracture surface, showing pull-outs during crack formation.The size of the pulled-out holes represents the formed cluster sizes.This morphology resulted in a relatively brittle composite structure.For Fe 3 O 4 the distribution is less regular in both the micro and the macro scales (Fig. 2e), but notable is the lack of separation between matrix and filler.This indicates that Fe 3 O 4 has a good compatibility with PBSA with used processing methods.Nanofillers are more likely to agglomerate at higher concentrations which can further impact polymer crystallinity and molecular mobility 30 .The strong agglomeration of nanoparticles is visible for both highly loaded compositions (Fig. 2c and  f) as large pulled-out cavities.It should be noted that both compositions reached a very high packing density.Micrographs at magnifications (1 000-10 000 ×) are available in the supplemental material (Figures S1-S10).
(5) Ŵ = To complement SEM results and show the exceptional distribution of nanoparticles SEM-EDS mapping was performed (Fig. 3).The most notable observation is that at 4 vol.%loading nanoparticles show almost no agglomeration (Fig. 3a and c).At a high loading the polymer matrix still retains a continuous network and effectively covers nanoparticles in the composite structure, even with increased filler agglomeration.
Figure 4 illustrates the tensile properties of the nanocomposites as a function of filler volume concentration: elastic modulus (Fig. 4a), yield strength (Fig. 4b), equivalent to ultimate stress for our samples, and yield strain (Fig. 4c).The introduction of fillers elevated the elastic modulus of the composites (Fig. 4a), reaching a peak at 1132 ± 234 MPa for 11 vol.%Fe 3 O 4 (20 nm).With increasing filler content, the composites transition to a more elastic yet brittle state.The NiFe 2 O 4 and Fe 3 O 4 (20 nm) fillers showed the most consistent and notable enhancement of elastic modulus.The yield strength (Fig. 4b) of the composites remained at the level of the neat system for most fillers with loadings up to around 10 vol.%.For Fe 3 O 4 (20 nm) a more pronounced drop in yield strength was observed.
As described previously 31 , PBSA undergoes a completely plastic failure characterized by extensive necking post-yield due to the orientation of the polymer chains; fractures occur at exceedingly high strains (elongation at break up to 250%).This orientation effect is maintained solely in composites with minimal filler concentrations, exemplified by the elongation at break values of 106.13 ± 59.39% for 4 vol.%Fe 2 O 3 , and 138.25 ± 16.93% and 81.70 ± 11.44% for 4 and 6 vol.%Fe 3 O 4 100 nm, respectively.Particle agglomeration serves to create stress concentrators, thereby reducing the overall ultimate strain by promoting crack initiation and facilitating crack propagation 32 .The agglomeration is related to filler characteristics, concentration, size, and dispersion.The likelihood of agglomeration and consequent embrittlement correlates with the average particle size of the fillers; composites filled with TiO 2 (5 nm) were excessively brittle to be molded into dog-bone-shaped specimens at higher filler loadings.This is corroborated by the reduced yield strains observed in NiFe 2 O 4 and Fe 3 O 4 (20 nm) composites (Fig. 4c).Beyond the yield point (Fig. 4c), all composites, with the exception of those previously mentioned, fractured at the onset of necking.showed superior volume-specific conductivities, whereas NiFe 2 O 4 and 20 nm Fe 3 O 4 particles had little to no conductivity improvement.The difference in filler conductivities f and the matrix conductivity m , has a negligible effect on the composite conductivities at low interconnectivities of filler particles.The geometry and size of the particles can also affect conductivity, by introducing lower-conductivity interfaces within the composites due to a difference in the surface area.It is expected that larger particles would have a higher influence on the thermal conductivity, however, in this case TiO 2 particles are an order of magnitude smaller than Fe 2 O 3 particles, yet they exhibit a similar effect on the thermal properties.Table 2 provides a comparative analysis of various thermally conductive composites, detailing similar filler concentrations reported in the literature alongside the composites investigated in this work.
To evaluate how well certain mathematical models can predict the heat-related properties of composite materials, we used a variety of models represented in the literature.These models were applied to data that we collected at a temperature of 25 °C.Table 3 provides a detailed presentation of the mathematical equations for the models.
The Maxwell model is one of the most used models [40][41][42][43] , also known as Maxwell-Garnett 40 and Maxwell-Eucken 41 , it parallels the Hashin-Strickman lower bounds and is prominent in describing thermal conductivity in randomly dispersed spherical non-interacting particles in a continuous polymer matrix.It assumes a continuous temperature profile at the filler surface, which proves effective at a macroscopic level.It becomes inadequate at significantly smaller nanoscales, where the assumption fails because thermal energy carriers, whether electrons or phonons, are scattered at the interface 40 .The Hatta model 44 , based on Eshelby's modified equivalent inclusion model model 47 , is mathematically equivalent to the Maxwell model.However, since their fitting curves coincide, only the Maxwell model is considered in further analysis.
The Bruggeman model was developed using different assumptions for permeability and field strength compared to the Maxwell model 40,43 .The model assumes that a composite material can be constructed incrementally by introducing infinitesimal changes to an already existing material.Thus, this approach is also called the differential effective medium theory.The Bruggeman model offers an advantage by taking into account interactions among randomly distributed fillers, and it can be extended to multi-component systems 45 .It also provides highly accurate predictions for composites with high volume fractions of fillers.The Lewis-Nielsen model is a semi-empirical model that considers the geometric features (shape, orientation and degree of packing) of filler particles 43,45,46 .Two additional parameters are introduced to the model: A -a constant related to the generalized Einstein coefficient k E , and V fmax -a maximum packing fraction of filler particles.The values of A and V fmax for different shapes and arrangements of particles are given in the reference 45 .In the current study for spherical particles, A = 1.5 and V fmax = 0.6 (body-centered cubic arrangement) are used in calculations.
The Agari model offers an alternative method for analyzing thermal conductivity 40,41 .This empirical model considers the dispersion state of the filler and the matrix structure by introducing two parameters, C 1 and C 2 .The first coefficient, C 1 , is related to the crystallinity and crystalline dimension of a polymer, and, the second one, C 2 , is the free factor, which indicates the ability of forming a heat conductive network for fillers.Therefore, C 2 would significantly change with an increase of filler content in composites.The empirical Agari model, reflecting the actual structure of composites, effectively fits thermal conductivity data in polymer composites filled with both micro-and nanoparticles 42,46 .
Figure 6 displays how the thermal conductivity data of PBSA composites align with the models listed in Table 3.The analysis uses a thermal conductivity value of 0.209 Wm −1 K −1 for the polymer matrix, and varying conductivities for fillers (in Wm −1 K −1 ): 8.4 (TiO 2 ), 10 (NiFe 2 O 4 ), 6 (Fe 2 O 3 ), and 5 (Fe 3 O 4 ).When approximating with the Agari model, it was assumed that the structure of PBSA is not affected by the presence of metal oxide particles, so C 1 is taken as unity.Best fits for the Agari model were obtained with the parameter C 2 equal to 0.7, 0.5, 0.9, 0.7, and 0.5 for composites filled with TiO 2 , NiFe 2 O 4 , Fe 2 O 3 , Fe 3 O 4 (100 nm), and Fe 3 O 4 (20 nm), respectively.According to the C 2 formulation, this parameter indicates the ease of creating thermally conductive filler networks specific to each polymer/filler system.The higher the C 2 , the more efficient thermal conductivity appears.2]46 A comparison of experimental thermal conductivity with the considered theoretical and empirical models shows that all approaches are able to reasonably predict the thermal conductivity of composites at low filler content and up to 20 vol.%.At higher filler loadings, the Bruggeman and Agari models provide better prediction due to their ability to account for interactions between fillers.
To determine how temperature affects thermal conductivity of the composites, we calculated the activation energies E a (Table 4 and Figure   www.nature.com/scientificreports/behavior is explained by the formation of a percolated network.For spherical nanofillers percolation happens when a sufficient packing factor is reached to permit the electrical contact of particles throughout the composite.Gurland has estimated that the onset of percolation starts at approximately 30-40 vol.% of conductive filler 51 .This somewhat aligns with our results, but the difference could be attributed to filler shape and size, dispersion conditions, and the properties of the matrix 52 .The ferrimagnetic nature of Fe 3 O 4 leads it to magnetically overpower dispersion forces, resulting in agglomeration.Consequently, this forms clusters that likely behave as  www.nature.com/scientificreports/larger, anisodiametric inclusions, thereby lowering the percolation threshold.The frequency independence at a certain critical frequency is a manifestation of the universal dielectric response and Jonscher's law 53 . The frequency-dependent permittivity of heterogeneous composites (Fig. 8) is largely determined by two key phenomena.The first is the Maxwell-Wagner-Sillars 54 effect, describing space charge polarization in dielectric composites.The second is Koops theory 55 , which explains semiconductor-filled composites as a system of conductive grains divided by resistive grain boundaries, focusing on the accumulation of charge at these boundaries.The high dielectric permittivity values in the Fe 3 O 4 composites at low frequencies could be attributed   56 and Li-Ni-Zn ferrites 57 .Fe 3 O 4 composites also exhibit increased dielectric permittivity due to charge hopping between the Fe +2 and Fe +3 ions at lower frequencies as described in our previous paper 14 .
With the presence of charge carriers this phenomenon is intensified at the highest concentrations, where conductive paths are established throughout the entire composite.In other metal oxide composites, space charge polarization occurs at lower frequencies.Above the relaxation frequencies of these space charges, the dielectric response becomes uniform and frequency-independent, transitioning to a state dominated by the matrix.In this state, the response is primarily controlled by the relaxation processes of the polymer 58 .

EMI shielding properties
To assess the potential of these composites for EMI shielding applications, we evaluated their shielding efficiency (SE) across various frequencies (Table 5).This was done using Eqs.( 5)- (7)  5), is generated by analog systems, electrical devices, power transmission equipment and others.This type of EMI is typically produced through conduction.Radio frequency interference is defined as EMI at frequencies from 20 kHz to optical wavelengths (shaded orange in Table 5).It is caused by radiative sources, such as wireless and radio communications systems, processors and microcontrollers, and high frequency equipment 59,60 .
S12) from Arrhenius plots (FigureS11) of the thermal conductivity measurements at 25, 35, 45 °C.Thermal activation energy refers to the energy required to overcome a thermally activated

Table 1 .
Prepared PBSA nanocomposites with experimental and theoretical density values, and extrapolated filler concentrations.

Table 2 .
Overview of various thermally conductive polymer composites documented in the literature.

Table 3 .
Models used for thermal conductivity fitting.
to space charge polarization.Notably, different authors have described this phenomena in other ferrite materials such as Zn-Cu ferrites Vol:.(1234567890) Scientific Reports | (2024) 14:13629 | https://doi.org/10.1038/s41598-024-64426-5www.nature.com/scientificreports/ , which consider losses due to both dielectric and conductive effects.The SE decreases as the field frequency increases.SE values were notably high at 0.01 Hz, reaching 77 dB for the 25 vol.% and 84 dB for the 40 vol.%Fe 3 O 4 (100 nm) loadings.However, as the frequency increased, there was a significant decrease in SE values, dropping to 9 dB and 11 dB for 25 vol.% and 40 vol.%composites, respectively.The high SE values of Fe 3 O 4 (100 nm) are caused by the formation of a conductive particle network.This network enhances electrical conductivity and dielectric losses, reflecting or effectively dissipating the EM field as heat.Fe 2 O 3 showed the second highest SE at the lowest frequencies.Notably, these SE values were inversely proportional to the filler concentration up to 10 Hz.This trend is associated with the dielectric relaxation processes that were discussed previously.The 4 vol.%Fe 2 O 3 composite is notable as it provides adequate EM shielding at a very low filler content, while simultaneously maintaining its non-conductive nature.It is important to evaluate the SE across different parts of the EM spectrum, as it varies depending on the specific application.Commonly, EMI in the very-low frequency or audio frequency, from direct-current non-alternating to 20 kHz alternating (shaded blue in Table

Table 5 .
Calculated EMI SE of the composite materials.